A Theory of Child Marriage

Zaki Wahhaj∗

University of Kent

January 2014

Abstract

The practice of early marriage for women remains prevalent in developing countries around the worldtoday, and is associated with various health risks, including infant mortality and maternal mortality as aconsequence of early pregnancy. This paper develops a dynamic model of a marriage market to explorewhether a norm of early marriage for women can prevail in the absence of any intrinsic preference foryoung brides. We show that if a certain desirable female attribute, relevant for the gains from marriage,is only noisily observed before a marriage is contracted, then its prevalence declines in each age cohortwith time spent on the marriage market; thus, age can signal poorer quality and require higher marriagepayments. In this situation, we show that interventions that increase the opportunity cost of earlymarriage —such as increased access to secondary schooling or adolescent development programmes —cantrigger a virtuous cycle of marriage postponement even if the original intervention is not suffi cient, initself, to persuade all women to turn down offers of early marriage.

∗Email: z.wahhaj@kent.ac.uk. Address: School of Economics, Keynes College, University of Kent, Canterbury CT2 7NP,United Kingdom.

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1 Introduction

Child marriage and the marriage of young adolescents remains prevalent in many parts of the world despite

repeated efforts by national governments and international development agencies to discourage and end the

practice. According to the State of World Population Report 2005, 48 per cent of women in Southern

Asia, and 42 per cent of women in Africa in the age group 15-24 years had married before reaching the

age of 18 (UNFPA 2005). Child marriage is likely to lead to early pregnancy and associated health risks

for the mother and the child; and force the marriage partners, especially the bride, to terminate schooling

prematurely. (Unicef 2001; Unicef 2005).

In recent years, international organisations and NGO’s have renewed efforts to discourage child marriage,

through interventions that raise awareness about its negative consequences, that provide parents incentives

to postpone marriage for their children, and that provide adolescents new opportunities to acquire skills and

alternatives to a traditional path of early marriage and early motherhood. Notable examples include Brac’s

Adolescent Development Programme in Bangladesh, which provides livelihood training courses, education

to raise awareness on social and health issues, and clubs to foster socialisation and discussion among peers;

and the Berhane Hewan project in Amhara, Ethiopia, a joint initiative between the New York based Popu-

lation Council and the Amhara regional government, which uses community dialogue, and simple incentives

involving school supplies to encourage delayed marriage and longer stay in school for girls. The World Bank,

the UK Department for International Development, and the Nike Foundation are providing support to a

number of similar projects around the world.

While parents and adolescents respond to incentives for delaying marriage, doing so appears to carry a

penalty on the marriage market: they are required to make higher marriage payments, a higher dowry, at

the time of marriage (Amin and Bhajracharya, 2011; Field and Ambrus, 2008).

In his well-known study of marriage patterns across the world, Jack Goody has highlighted a number of

reasons why young brides are preferred in traditional societies: they have a longer period of fertility before

them; and they are more likely to be obedient and docile, necessary qualities to learn and accept the rules and

ways of her new household (Goody, 1990). Relatedly, Dixon (1971) attributed the historic practice of early

marriage in China, India, Japan and Arabia to the prevalence of ‘clans and lineages’which gave economic

and social support to newly married couples, as well as pressures to produce children for strengthening and

sustaining the clan. By contrast, the traditional emphasis on individual responsibility in ‘Western family

systems’meant that newly married couples were expected to be able to provide for themselves and their

children, which ‘necessarily causes marital delays while the potential bride and groom acquire the needed

skills, resources and maturity to manage an independent household’(Dixon 1971).

However, these explanations of early marriage do not, in themselves, explain why the age of marriage

for women has remained low in some societies despite declines in fertility, and evolution towards a nuclear

household model. Questions of this nature require an understanding — not just about the preferences,

opportunities and constraints faced by parents and households that practise child marriage —but about the

’marriage market equilibrium’, the conditions under which the practice of child marriage may be collectively

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sustained in a society. Similarly, the eventual consequences of any large-scale programme which aim to

discourage child marriage will depend —not only on how individual households respond to these incentives

—but on the equilibrium effects of these interventions.

In this paper, we develop a dynamic model of a marriage market to obtain insights about the practice

of child marriage. We show that even if an inherent preference for young brides is absent among potential

grooms and their families, a scenario where they all seek young brides is a stable equilibrium; and under

certain conditions, the unique equilibrium.

The equilibrium preference for young brides is driven by a kind of informational asymmetry. We assume

that a certain attribute considered desirable in a bride is unobservable except for a noisy signal received

when a match has been arranged. In equilibrium, the prevalence of this attribute declines in each age cohort

with time spent on the marriage market; thus, age of the bride can signal quality. Young potential brides,

aware that they will be perceived as being of poorer quality the longer they remain on the marriage market

would, therefore, have an incentive to accept an offer of marriage sooner rather than later.

Large-scale interventions of the kind discussed earlier provide adolescent girls with opportunities other

than marriage. If these opportunities are suffi ciently attractive for at least some adolescent (young) girls

to turn down offers of marriage, then this reduces the correlation between the age of a potential bride and

her quality. In turn, this would provide other women incentives to turn down offers of marriage, which

further reduces the correlation between the age of a potential bride and her quality. Thus, expanding non-

marriage related opportunities for adolescents can trigger a virtuous cycle of marriage postponement even if

the original intervention is not suffi cient, in itself, to persuade all women (and their families) to turn down

offers of marriage.

The fact that an intervention targeted at adolescent girls can make it more and more attractive for

future cohorts to postpone marriage means that the long-term impact of such interventions on marriage and

subsequent life choices may well exceed the immediate impact.

A recent literature on marriage practices around the world has explored the possibility that certain,

potentially harmful, practices can be maintained as self-sustaining equilibria. Mackie (1996) applies the

concept of Shelling’s focal point to explain why the practice of footbinding in China had survived over a

thousand years, and how it came to an end very quickly at the beginning of the 20th century. Mackie

(2000) and Mackie and LeJeune (2009) applies the same game-theoretic framework to the practice of female

circumcision and argues that it may be undergoing a similar transition in West Africa now, at the beginning

of the 21st century. In a similar spirit, this paper investigates whether the practice of child marriage can be

sustained as a self-sustaining equilibrium in the absence of intrinsic preferences for young brides; and if so,

under what conditions could such an equilibrium unravel.

Relatedly, Bidner and Eswaran (2012) provides an explanation for the emergence of the caste system in

India that also accounts for a number of marriage-related practices common to the Indian context, including

child marriage. They develop a model of production and exchange in which complementarity of skills between

the husband and wife in household production gives rise to a preference for endogamy within groups that are

engaged in distinct occupations. The practice of child marriage (as well as arranged marriages, etc.) reduce

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the risk of group members coming into contact with —and mating with —members of other (caste) groups.

As the practice of child marriage exists in contexts where caste and caste-like institutions do not prevail —

such as muslim communities in South Asia, and in Africa —the phenomenon also calls for other explanatory

factors, such as the mechanism provided in this paper.

2 A Model of Marriage Timing

In this section, we develop a dynamic model of a marriage market to explore the conditions under which

child marriage may be sustained in equilibrium, and conditions under which the practice may unravel.

The theoretical model presented in this paper follows the literature on dynamic search and matching

initiated by Diamond and Maskin (1979). The framework has previously been used to investigate a number

of marriage-related phenomena. For example, Bergstrom and Bangoli (1993) develop a matching model to

explain the age gap between husbands and wives. Anderson (2007) investigates the effect of population

growth on rising dowry prices. Sautmann (2011) provides a characterisation of the marriage payoff functions

which would lead to commonly observed marrage age patterns around the world, including positive assortative

matching in age and a positive age gap between grooms and brides.

Since our key interest is the question of marriage timing, we make certain simplifying assumptions vis-

a-vis the existing literature on marriage markets; assumptions which, we will argue, are reasonable given

contemporary marriage patterns in South Asia. In contrast to Sautmann’s paper, we abstract away from any

intrinsic age-related preferences on the marriage market and focus, instead, on the imperfect observability

of the characteristics of potential marriage partners to explain the phenomenon of early marriage.

In Arguing with the Crocodile, Sarah White provides a detailed ethnographic account of marriage practices

in rural Bangladesh. She notes that it is customary for the groom’s family to initiate contact with the

bride’s family, and that this contact is made through a ‘matchmaker’who provides a communication line,

and facilitates negotiations, between the two parties (White 1992). She notes that "the different parties

manouvre and fight their own corners, aiming to achieve the best bargain they can" which suggests that

bargaining is a key element of the negotiation process. Moreover "the many interests involved in the making

of a marriage undoubtedly color the evidence given. Mismatches occur not only through lack of information,

but also through deliberate deception", which suggests that each party has limited information about the

potential match, and that they may choose not to disclose information about their daughter or son that may

be regarded as a defect on the marriage market. Therefore, "... attention ... is often focused primarily on

the wealth of the household, and the amount of dowry which is demanded or offered" (White 1992).

In line with this description, we model the matching process in the marriage market, as one that is initiated

by the potential groom (or his family), via a match-maker, as discussed in the next section. Furthermore, we

assume that both the potential bride and groom have unobserved characteristics relevant for the marriage

which may be revealed through a ‘background check or inferred from observable characteristics such as

their age. The marriage transfer is agreed upon through bargaining between the two parties. There is no

remarriage market; i.e. once a bride and groom are married, they have no possibility of re-entering the

marriage market to seek a new partner. And women remain on the marriage market —i.e. are considered

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eligible for marriage —during a finite number of years. These last two assumptions seem justified given that,

in South Asia, the incidence of separation and divorce remains very low (Dommaraju and Jones, 2011) and

that most women marry by their late twenties (Mensch, Singh and Casterline, 2005).

2.1 Description of the Marriage Market

The marriage market consists of four types of individuals: ‘young’men, ‘young’women, ‘older’men and

‘older’women. The number of individuals of each type at time t are given by nm1 (t), nm2 (t), nf1 (t) and

nf2 (t) respectively.

Two individuals of the same type are exactly alike in terms of their observable characteristics. Women

have an unobservable characteristic which we call ‘moral character’, which may be either ‘good’or ‘bad’.

Women in the population have ‘bad moral character’with probability εf ∈ (0, 1), a constant that is exoge-nously given.

The utility from marriage depends on the marital transfers involved and the moral character of the

partners. If a marriage between a man and a woman involves a net pre-marital transfer τ from the bride

to the groom, and the probability that the bride has ‘bad moral character’ is εf , then we represent the

utility to the groom by the function um (τ , εf ). Similarly, if the probability that the groom has ‘bad moral

character’is εm, then the utility to the bride is given by uf (τ , εm). Note that men are assumed not to have

any intrinsic preference between ‘young’and ‘older’brides; and women are assumed not to have any intrinsic

preference between ‘young’and ‘older’grooms.

The matching process between prospective grooms and prospective brides operates according to the

following sequence of events:

1. Men state a preference as to whether they would like to be matched with a ‘young’woman or an ‘older’

woman. A search is then initiated by the matchmaker.

2. The probability that the matchmaker finds a match for a man seeking a young bride is given by the

function µ(nf1θnm

)where θ is the proportion of all men who state a preference for a young bride at

stage 1, and nm = nm1+nm2. Similarly, the probability that the matchmaker finds a match for a man

seeking an older bride is given by µ(

nf2(1−θ)nm

).

3. If a match has been found, the prospective bride and groom decide whether they will enter into a

discussion regarding marriage.

4. If marriage discussions are initiated, a ‘background check’is performed on the prospective bride. For

the sake of simplicity, we assume no such technology is available to perform a ‘background check’on

the prospective groom.

5. The background check can have two possible outcomes. It may reveal nothing or it may reveal that

the prospective bride has ‘bad moral character’. If a prospective bride has ‘bad moral character’, then

the background check reveals her true character with probability π ∈ (0, 1).

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6. The two sides decide whether the marriage should go ahead. If they decide the marriage will not take

place, they remain single for the remainder of the period. The utility levels obtained from being single

during one period are denoted by um and uf for men and women respectively.

7. If they decide to marry, then a bargaining process determines the level of pre-marital transfers, if any.

Specifically, if the probability that the prospective groom and bride have ‘bad’moral character are

given by εm and εf respectively, and the utility to the man and woman from their outside options

are vm and vf respectively, then the net marital transfers (from the bride to the groom) agreed upon

through bargaining is given by τ = ξ (vm, vf , εm, εf ).

When any individual marries, he or she leaves the marriage market with no possibility of re-entry. Young

women and young men who fail to/choose not to marry in a particular period, re-enter the marriage market

as, respectively, older women and older men in the next period. Older women and older men who fail

to/choose not to marry in a particular period must remain single thereafter (i.e. they leave the marriage

market). In each period, a new cohort of y young women and y young men reach marriageable age. To

simplify the dynamics, we further assume that young men prefer not to marry (perhaps because they do not

have the means to support a family till they become ‘older’).

We make the following assumptions about the functions um (.), uf (.), µ (.) and ξ (.).

Assumption 1 um (τ , εf ) is strictly increasing in τ and strictly decreasing in εf .

Assumption 2 uf (τ , εm) is strictly decreasing in τ and strictly decreasing in εm.

Assumption 3 µ (.) is strictly increasing with a range of [0, 1] and µ (0) = 0.

Assumption 4 θµ(xθ

)is increasing in θ for all x.

Assumption 5 ξ (vm, vf , εm, εf ) is (weakly) increasing in vm, decreasing in vf , decreasing in εm and in-

creasing in εf .

Assumption 6 um + uf > um (τ , 1) + uf (τ , εm) and um + uf > um (τ , εf ) + uf (τ , 1) for all τ .

Assumption 7 um < um (τ , εf ) and uf < uf (τ , εm) for some τ .

Assumption 8 um (ξ (vm, vf , εm, εf ) , εf ) is decreasing in εf and uf (ξ (vm, vf , εm, εf ) , εm) is decreasing in

εm.

Assumption 6 implies that if either the prospective bride or groom is found to have bad moral character

(with no new information obtained regarding the other party), then there is no level of pre-marital transfers

such that both parties would prefer marriage to singlehood). By contrast, Assumption 7 implies that if no

new information is obtained regarding the prospective bride or groom, then there exits a level of pre-marital

transfers such that both parties would prefer marriage to singlehood.

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Assumption 4 is a restriction imposed on the function µ (.) which ensures that as the proportion of men

who state a preference for a particular type of bride (either ‘young’or ‘older’) increases, so does the number

of men who are matched with that type of bride.

Assumption 8 implies that as the reputation of a prospective partner improves, so does the utility from

marrying him or her, even after taking into account that a partner with a better reputation will require a

higher net marriage transfer.

Future utility is discounted by a factor β ∈ (0, 1) per period. For the sake of simplicity, we assume thatmen and women are infinitely lived (although they remain on the marriage market for only a finite number

of periods).

We denote by λ1 and λ2, respectively, the probability of finding a match for young women and older

women on the marriage market. By construction, we must have

λ1nf1 = µ

(nf1θnm

)θnm (1)

λ2nf2 = µ

(nf2

(1− θ)nm

)(1− θ)nm (2)

2.2 Solving the Model

Whether men prefer young brides or older brides will depend on the ‘reputation’ (i.e. the probability of

having bad moral characer) of each cohort.

Background Checks: Let us denote by εf1 and εf2 the probability of ‘bad moral character’among

young and older women respectively before any background check has been performed; and, by εf1 and εf2,

the probability of ‘bad moral character’ among young and older women respectively after a background

check has been performed and the check has revealed nothing.

As there is no information available on the moral character of young women when they first enter the

marriage market, we must have εf1 = εf . For older women, the probability of bad moral character (before a

background check has been performed) depends on the likelihood that they were matched with a man when

young. Specifically, using Bayes’rule, we obtain

εf2 = Pr (bad|older)

=Pr (older|bad) Pr (bad)

Pr (older)

=[(1− λ1) + λ1π] εf(1− λ1) + λ1πεf

(3)

From (3), it is evident that if λ1 > 0 and εf > 0, then εf2 >[(1−λ1)+λ1π]εf(1−λ1)+λ1π = εf . Therefore, if there are

some women who are getting married when young, the probability of bad moral character is higher among

older woman than among young women.

Given that a background check detects ‘bad moral character’only with some probability π < 1, a woman

for whom such a check has revealed nothing, will still be assigned a positive probability of having a bad

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moral character. We can determine this probability for young and older women using Bayes’rule as follows:

εf1 = Pr (bad|young+nothing)

=Pr (young+nothing|bad) Pr (bad)

Pr (young+nothing)

=(1− π) εf1

(1− εf1) + (1− π) εf1(4)

<(1− π) εf1

(1− π) (1− εf1) + (1− π) εf1< εf1

εf2 = Pr (bad|mature+nothing)

=Pr (mature+nothing|bad) Pr (bad)

Pr (mature+nothing)

=(1− π) εf2

(1− εf2) + (1− π) εf2(5)

<(1− π) εf2

(1− π) (1− εf2) + (1− π) εf2< εf2

So, when a background check on a woman reveals nothing, the probability that she has bad moral

character declines but remains positive, in case of both young women and older women.

In summary, we have established the following results.

Summary 1 εf1 = εf and εf2 > εf

εf1 < εf1 and εf2 < εf2

εf2 > εf1 and εf2 > εf1

Marriage Transfers: At stage 7 of the sequence of events in the game, matched men and women

bargain over the level of pre-marital transfers. It should be evident that all young women have the same

outside option as they are indistinguishable from one another on the marriage market. Similarly, all older

men have the same outside option and all older women have the same outside option.

Therefore, the level of premarital transfers agreed upon through a process of bargaining in any marriage

involving a young woman (for whom a background check has revealed nothing) and an older man should be

the same. We denote this level of transfers by τ1.

Similarly, the level of premarital transfers agreed upon through a process of bargaining in any marriage

involving an older woman (for whom a background check has revealed nothing) and an older man should be

the same. We denote this level of transfers by τ22.

Outside Options: Let us denote by vf1 and vf2 respectively the outside options of young women and

older women on the marriage market. An older woman who does not marry in the current period will remain

single hereafter. Therefore, her outside option is given by

vf2 = ζuf

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where ζ = 11−β . A young woman who does not marry in the current period will re-enter the marriage market

as an older woman. She would need to consider not only whether she will find a partner when she is older,

but also whether a second background check would reveal bad moral character. Therefore, her probability

of marrying as an older bride is given by λ′2 = λ2 (1− πεf2). Therefore, her outside option satisfies thefollowing equation:

vf1 = uf + βζ{λ′2uf (τ2, εf2) +

(1− λ′2

)uf}

={1 + βζ

(1− λ′2

)}uf + βζλ

′2uf (τ2, εf2) (6)

Let us denote by vm the outside option of older men on the marriage market (Since young men do not

marry by assumption, we need not consider their outside options for the analysis). An older man who does

not marry in the current period will remain single hereafter. Therefore, his outside option is given by

vm = ζum

If an older man states a preference for a young bride, then a marriage with a young bride takes place

with probability µ(nf1θnm

)(1− εf1). From this outcome, he receives a continuation utility of ζum (τ1, εf1).

Therefore, the expected utility to an older man from stating a preference for a young bride is given by

U1 = µ

(nf1θnm

)(1− εf1) ζum (τ1, εf1) +

[1− µ

(nf1θnm

)(1− εf1)

]vm (7)

If he states a preference for an older bride, then a marriage with an older bride takes place with probability

µ(nf2θnm

)(1− εf2). From this outcome, he receives a continuation utility of ζum (τ2, εf2). Therefore, the

expected utility to an older man from stating a preference for an older bride is given by

U2 = µ

(nf2θnm

)(1− εf2) ζum (τ2, εf2) +

[1− µ

(nf2θnm

)(1− εf2)

]vm (8)

Bargaining: Given the outside options of older men, young women and older women on the marriage

market, we can write the marital transfers that would occur in the different types of marriages as follows:

τ1 = ξ (vm, vf1, εf1) (9)

τ2 = ξ (vm, vf2, εf2) (10)

2.3 Equilibrium in the Marriage Market

In any period, the state of the marriage market can be fully described by the number of individuals in each

age-gender group. By assumption, there are y young men and y young women in the marriage market in

each period. As we have assumed that young men do not marry, there are also y older men in the marriage

market each period. Therefore, the only quantity which can potentially vary over time is the number of older

women in the marriage market.

If a fraction λ1 of young women are matched with men in some period, then the number of older women

in the marriage market in the next period is given by

nf2 = [1− λ1 + λ1επ] y (11)

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The logic behind (11) is as follows: if a fraction λ1 of young women are matched with men, then a fraction

λ1επ will be discovered to have ‘bad moral character’. Therefore, those who remain on the marriage market

in the next period would include those who were not matched, numbering (1− λ1) y1 and those who werefound to have ‘bad moral character’, numbering λ1επy. By construction, επ < 1. Therefore, it follows from

(11) that nf2 is decreasing in λ1.

Letting nf1 = nm2 = y in (1), we obtain

λ1y = µ

(y

θy

)θy

Then, by Assumption 4, λ1 is increasing in θ. It follows that nf2 is decreasing in θ. Therefore, the number

of older women in the marriage market is declining in the fraction of men who (in the preceding period)

state a preference for young brides.

Thus, we can represent the state of the marriage market with a single variable, λ1 —the probability that

a young woman will be matched with a potential groom —. which, in turn, is determined by θ, the proportion

of men who state a preference for young brides.

We are now in a position to characterise equilibria in the marriage market. Let us denote by θt the value

of θ in period t. From equation (7), we can see that the expected utility to older men in some period t, from

stating a preference for young brides, U1, depends on θt. The outside option of young brides, and therefore

the equilibrium marriage payments depend on the marriage prospects of older women in the next period.

Thus, the value of U1 in period t also depends on θt+1. All other terms on the right-hand side of (7) are

exogenously determined. Therefore, we can write the expected utility from stating a preference for a young

bride as a function U1 (θt, θt+1).

From equation (8), we can see that the expected utility to older men in period t from stating a preference

for older brides, U2, also depends on θt. The reputation of older brides, and therefore the utility from

marrying older brides depends on the proportion of young women who were married in the previous period.

Thus, the value of U2 in period t also depends on θt−1. All other terms on the right-hand side of (8) are

exogenously determined. Therefore, we can write the expected utility from stating a preference for an older

bride as a function U2 (θt, θt−1). It is straightforward to establish the following results.

Lemma 1 (i) Under Assumption 3, U1 (θt, θt+1) is strictly decreasing in θt and U2 (θt, θt−1) is strictly

increasing in θt.

(ii) Under Assumptions 1 and 5, U1 (θt, θt+1) is strictly increasing in θt+1

(iii) Under Assumption 8, U2 (θt, θt−1) is strictly decreasing in θt−1

Proof. See the Appendix.

The first part of Lemma 1 follows directly from the properties of the functions um (., .) and µ (.). The

second part of Lemma 1 has the following intuition: as the future marriage prospects of young women

improve, they will require higher net marriage payments for marrying as young brides. Therefore, the

expected utility to men from seeking young brides will decline. The third part of Lemma 1 has the following

intuition: if a large fraction of men sought young brides in the preceding period, then the reputation of older

women, and therefore the expected utility from seeking an older bride, is low in the current period.

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0 1

U2

U1

decreasingin θt­1

increasing θt

increasingin θt+1

If, in equilibrium, θt ∈ (0, 1), then we must have U1 (θt, θt+1) = U2 (θt, θt−1). If not, some men would be

able to improve their expected utility by changing their age preference for prospective bride. From Lemma

1(i), we see that U1 (.) is monotonically decreasing, and U2 (.) is monotonically increasing, in θt. Therefore

there is, at most, one value of θ ∈ [0, 1] which satisfies the preceding equation, as in Figure 1.

Similarly, we can reason that if θt = 0, we must have U1 (0, θt+1) ≤ U2 (0, θt−1) and if θt = 1, then

U1 (1, θt+1) ≥ U2 (1, θt−1). These arguments enable us to compute the equilibrium value of θt whenever θt−1

and θt+1 are known. For this purpose, let us define function I : [0, 1]× [0, 1] −→ [0, 1] as follows:

Definition 2

I (θt−1, θt+1) =θ if U1 (θ, θt+1) = U2 (θ, θt−1) for some θ ∈ [0, 1]0 if U1 (θ, θt+1) < U2 (θ, θt−1) for each θ ∈ [0, 1]1 if U1 (θ, θt+1) > U2 (θ, θt−1) for each θ ∈ [0, 1]

Using this definition, we can establish the following result.

Proposition 1 Given an initial value θ0, the sequence {θτ}∞τ=1 constitutes a Perfect Baysian Equilibriun ifand only if θt = I (θt−1, θt+1) for t = 1, 2, ...

Proposition 1 provides us a convenient characterisation of all possible equilibria in the marriage market

and the different ways in which the marriage age for women may evolve over time when all individuals in

the marriage market are choosing their best response.

11

0 1

increasing θt

I(θ,θ)

steady­state values

3 Steady-State Equilibria

Suppose there is some θ ∈ [0, 1] which satisfies the following equation:

I (θ, θ) = θ

In this case, the strategy profile where a fraction θ of men seek young brides in every period constitutes

an equilibrium (when the initial value is also θ).

Lemma 2 There exists at least one steady-state value θ ∈ [0, 1]; i.e. I (θ, θ) = θ.

Proof. See the Appendix.

Figure 2 shows a plot of I (θ, θ) against θ. It follows from Lemma 1(i) that the curve is upward-sloping.

At θ = 0, it must lie on or above the 45-degree line and at θ = 1, it must lie on or below the 45-degree

line. From Lemma 2, we know that it must cross the 45-degree line at least once. Each value of θ where the

curve crosses the 45-degree line constitutes a steady-state equilibrium. Below we provide two examples of

potential steady-state equilibria.

1. Suppose I (1, 1) = 1. Then, there exists a steady-state equilibrium in which, in each period, all men

seek young brides.

The probability that a young woman will find a match is given by λ1 = µ(yy

)yy .

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Older women do not receive any offers of marriage, and λ2 = 0.

Therefore, vf1 = vf2 = ζuf ; i.e. both young and older women have the same outside option.

Older women have worse ‘reputation’than young women: εf2 > εf1.

Furthermore, we have

U1 = µ(yy

)(1− εf1) ζum (τ1, εf1) +

[1− µ

(yy

)(1− εf1)

]vm

U2 = µ(λ1y1

)(1− εf2) ζum (τ2, εf2) +

[1− µ

(λ1y1

)(1− εf2)

]vm

τ1 = ξ (vm, vf1, εf1)

τ2 = ξ (vm, vf2, εf2) (with εf2 being derived from λ1)

Since εf2 > εf1, we have τ1 < τ2

Since I (1, 1) = 1, we have U1 (1, 1) ≥ U2 (1, 1); i.e. the men obtain a higher expected utility from

seeking a young bride as opposed to an older bride.

From the point of view of potential grooms, young brides are harder to find and they are more ‘expen-

sive’. But if the reputational effect offsets these disadvantages, we have a ‘child marriage’equilibrium

as described above. Alternatively, we could have the following situation.

2. Suppose I (0, 0) = 0. Then, there exists a steady-state equilibrium in which, in each period, all men

seek older brides.

Young women do not receive any offers of marriage and λ1 = 0. Therefore, nf2 = y.

Older women receive offers of marriage with probability λ2 = µ(yy

)yy .

Therefore, vf1 = {1 + βζ (1− λ2)}uf + βζλ2uf (τ2, ε2) and vf2 = ζuf .

Since λ2 > 0, we have, vf1 > vf2; i.e. younger women have a better outside option than older women.

Older women have the same reputation as young women: εf2 = εf1

Furthermore, we have

U1 = µ(y1

)(1− εf1) ζum (τ1, εf1) +

[1− µ

(y1

)(1− εf1)

]vm

U2 = µ(yy

)(1− εf2) ζum (τ2, εf2) +

[1− µ

(yy

)(1− εf2)

]vm

τ1 = ξ (vm, vf1, εf1)

τ2 = ξ (vm, vf2, εf2)

Since vf1 > vf2, we obtain τ1 < τ2

Since I (0, 0) = 0, we have U2 (0, 0) ≥ U1 (0, 0); i.e. the men obtain a higher expected utility from

seeking an older bride as opposed to a young bride.

From the point of view of potential grooms, older brides are harder to find. But if the lower expense of

older brides (they require a lower net marital transfer) offsets this disadvantage, we have a delayed marriage

equilibrium.

13

For both examples, we obtain the result that young brides make a smaller net marital transfer than older

brides. Note that U2 (0, 0) = U1 (1, 1). In other words, the men receive the same expected utility in the

two potential steady-state equilibria. This is because, if θt−1 = 0, the reputation of older women in period

t is the same as that of young women; and, if θt+1 = 1, the outside option of young women in period t is

the same as that of older women. Therefore, from the point of view of potential grooms, older women in

the θ = 0 steady-state are ‘identical’—i.e. they have the same reputation and require the same net marital

transfers —as younger women in the θ = 1 steady-state.

There are, potentially, other steay-state equilibria where θ takes some fixed value between 0 and 1 in each

period. This would mean that a fixed proportion of men seek young brides and others seek older brides, and

the expected utility from the two choices are the same: U2 (θ, θ) = U1 (θ, θ). In all of these equilibria, young

women have a better outside-option and a better reputation than older women. Therefore, young brides

make a smaller net marital transfer than older brides.

4 Equilibria with Varying θ

In the preceding section, we considered equilibria where a fixed proportion of men sought young brides in

every period. Next, we consider a distinct class of equilibria; namely, those equilibria where this proportion

changes over time. Note that although θ would change period by period, the marriage market would be in

equilibrium in each period in the sense that all agents are choosing their best response, given the current state

of the market and their expectation about the future. Furthermore, it is noteworthy that the marriage market

can have such dynamics even though we have not, so far, introduced any external shocks or interventions.

The diffi culty of analysing this class of equilibria is that they are driven by expectations about future

values of θ. A marriage market where there is an expectation of wide prevalence of early marriage in the

future would have a different trajectory from another marriage market where there is an expectation of a

wide prevalence of late marriage. In other words, we can have multiple equilibria for the same initial value

of θ.

Fortunately, there is one simple refinement that dramatically reduces the number of potential equilibria.

The refinement is described in the following definition.

Definition 3 Given an initial value θ0, the sequence {θτ}∞τ=1 constitutes a monotonic increasing equilibriumif θt = I (θt−1, θt+1) and θt ≥ θt−1 for t = 1, 2, .... Given an initial value θ0, the sequence {θτ}∞τ=1 constitutesa monotonic decreasing equilibrium if θt = I (θt−1, θt+1) and θt ≤ θt−1 for t = 1, 2, ....

This refinement provides us considerable predictive power about how the age of marriage will evolve over

time, using just the initial value of θ0 and the fundamental parameters of the marriage market, as described

in the following proposition.

Proposition 2 (i) Given an initial value θ0, there exists a monotonic increasing equilibrium if and only if

there is a steady-state θ ∈ (θ0, 1]. The equilibrium sequence {θt}∞t=1 converges to a steady-state θ ∈ (θ0, 1].(ii) Given an initial value θ0, there exists a monotonic decreasing equilibrium if and only if there is a

steady-state θ ∈ [0, θ0). The equilibrium sequence {θt}∞t=1 converges to a steady-state θ ∈ [0, θ0).

14

Proof. See the Appendix.

We can take an alternative approach based on the assumption that individuals are naive about their beliefs

regarding the future marriage market; specifically, that their decisions in period t rely on the assumption that

θt+1 = θt−1. This assumption is unsatisfying in the sense that people are repeatedly wrong about their future

beliefs in the marriage market. However, as we shall show, their beliefs grow more and more accurate over

time as θ approaches a steady-state value; moreover, it generates a unique and easily computable equilibrium

path in the marriage market.

Definition 4 Given an initial value θ0, the sequence {θτ}∞τ=1 constitutes a Naive Expectations Equilibriumif, θt = I (θt−1, θt−1) for t = 1, 2, ...

Note that a ‘Naive Expectations’equilibrium does not satisfy the criteria for a Perfect Bayesian equilib-

rium. A steady-state value of θ satisfies the conditions for both the Perfect Bayesian equilibrium and the

‘Naive Expectations’equilibrium. The advantage of a ‘Naive Expectations’equilibrium is that the initial

value θ0 uniquely determines the equilibrium path, as described in the following proposition.

Proposition 3 Given an initial value θ0, there is a unique sequence {θτ}∞τ=1 that satisfies the criteria of a‘Naive Expectations’equilibrium. Furthermore,

(i) if I (θ0, θ0) > θ0, then {θτ}∞τ=1 is an increasing sequence which converges to the smallest steady-statein the interval (θ0, 1];

(ii) if I (θ0, θ0) < θ0, then {θτ}∞τ=1 is a decreasing sequence which converges to the largest steady-statein the interval [0, θ0);

(iii) if θ0 = I (θ0, θ0), then θτ = θs for τ = 1, 2, ....

Proof. See the Appendix.

5 Changing the Opportunity Cost of Early Marriage

Interventions that improve access to secondary schooling or provide other training opportunities for girls

can, potentially, raise their utility from singlehood. There are two channels through which it may do so.

If attending school and participating in adolescent development programmes are enjoyable activities then

such interventions would, in expectation, improve the quality of life for an adolescent. Attending school

can, potentially, also improve one’s ability to generate an income in later life, and consequently increase the

expected utility from adulthood.

Marriage can make it considerably more diffi cult to access these opportunities. Field and Ambrus (2008)

show that, in rural Bangladesh, age of marriage is a strong predictor for age of leaving school. Maertens

(2012) finds, for villages in rural India, that parents’perception of what is an acceptable age of marriage

is a significant predictior of their educational aspirations for their daughters, indicating that marriage and

further education are regarded as mutually exclusive choices. Stipend programmes, such as the Female

Secondary School Assistance Program (FSSAP) in Bangladesh, are often conditional on the pupil remaining

single (Asadullah and Chaudhury 2009). Therefore, it is reasonable to assume that women who opt to marry

15

while they are ‘young’forgo further educational opportunities. If so, improved access to secondary schooling

and adolescent development programmes would make singlehood more attractive, relative to marriage, for

young women.

If so, then some women may find it in their interest to ‘turn down offers of marriage’even if they are

matched with a potential groom when they are young; in other words, Assumption 7 —according to which

there is always some level of transfer between a potential bride and a potential groom such that both would

prefer marriage to singlehood —may not hold true.

The effect of these opportunities on the marriage decision of older women is more ambiguous. Schooling

can improve their outside options (e.g. access to the labour market, further education, or a different segment

of the marriage market). But schooling can also improve the total marriage surplus over which the bride

and the groom bargain. For the present analysis, we disregard this effect. This is equivalent to assuming

that, for older women on the marriage market, improved access to schooling and adolescent development

programmes do not affect the relative attractiveness of singlehood versus marriage.

In Section 5.2, we consider how improving opportunities of adolescent women — by improving access

to schooling, or other opportunities for acquiring skills and knowledge —would impact upon the marriage

marriage. We assume that women who do not marry young (whether because they were not matched with

a potential groom, or were found to have ‘bad moral character’, or turned down an offer of marriage) choose

to pursue schooling till they become ‘older’. Therefore, all ‘older’women on the marriage market would

have attended school; and so their level of education do not provide any additional information about their

‘moral character’.

To investigate the impact of programmes that improve the opportunities of adolescent women, we first

need to analyse how the refusal of offers of marriage by some young women can affect the marriage market

equilibrium. We do this in the next section.

5.1 Turning Down Offers of Marriage

Let us denote by uf1 the utility from singlehood when a woman is ‘young’and, by uf2, the utility from

singlehood when a woman is ‘older’. To capture situations where some, but not all, young women may

find it in their interest to turn down offers of marriage, we allow the utility from remaining single while

young to vary across individuals. This may be due to differentiated access to secondary schools or adolescent

development programmes, and differences in preferences. Specifically, let uf1 = y + ψ where y is a constant

common to all individuals and ψ is distributed across individuals according to the cummulative distribution

function F (.).

We can define x as the threshold value of the utility from singlehood at which a young woman is indifferent

between accepting and refusing an offer of marriage. Then, x must satisfy the following equation:

(1 + βζ)uf (τ1, εf1) = x+ βζ[(1− λ2)uf2 + λ2uf (τ2, εf2)

](12)

where τ1 solves um (τ1, εf1) = um. Therefore, for uf1 > x, a woman would, in effect, choose to delay

marriage. If F (x− y) = 1, then we obtain the situation described in Section 2.3. If F (x− y) < 1, some

16

matches involving young women will not lead to marriage even when the background check has not signalled

‘bad moral character’. This has two implications for the marriage of young women on the marriage market.

First, a potential groom is less likely to be matched with a young bride with whom a marriage contract can

be negotiated. The probability of marriage for men who state a preference for young brides will be given by

µ

(nf1θnm

)(1− εf1)F (x− y)

where θ, as before, is the proportion of men who seek young brides.

Second, if some women are postponing marriage when they are young, it provides an alternative reason

why older women may be single (other than the two reasons discussed above: not being matched with a

potential groom and being found to have ‘bad moral character’); and, consequently, the reputation of older

women on the marriage market would improve. Specifically, it is given as follows:

εf2 = Pr (bad|older)

=Pr (older|bad) Pr (bad)

Pr (older)

=⇒ εf2 =

εf [(1− λ1) + λ1π + ηλ1 (1− π)](1− εf ) [(1− λ1) + ηλ1] + εf [(1− λ1) + λ1π + ηλ1 (1− π)]

(13)

where η = 1− F (x− y) denotes the proportion of young women who turned down offers of marriage in thepreceding period. If η = 0, the expression in (13) is identical to that in (3). As η increases, εf2 declines; i.e.

there is a fall in the probability of ‘bad moral character’among older women on the marriage market. In

addition, the number of older women in the marriage market will go up with the proportion of young women

who declined offers of marriage in the preceding period (represented by η′ below):

nf2 = [1− λ1 + η′λ1επ] y (14)

Equations (13) and (12) together yield, for a given value θ, the threshold value x above which young women

turn down offers of marriage and η, the proportion of young women who do so.1 Let us denote these values

by the functions x (θ) and η (y, θ) respectively. (We suppress the parameter y in the function η (.) hereafter

in this section for ease of notation. The consequences of changing y are explored in the next section).

As per the reasoning above, the expected utility to potential grooms from a seeking young bride, in a

given period t, will depend on ηt, the proportion of young women who decline offers of marriage in that

period. On the other hand, the expected utility from seeking an older bride will depend on ηt−1, because

the proportion of young women who declined offers of marriage in the preceding period determines, as per

(13), the reputation of older women in the current period. We can therefore represent the expected utilities

by U1 (θt, θt+1, ηt) and U2(θt, θt−1, ηt−1

)respectively. Using these expected utility functions, we can define

a function I (.) akin to the function I (.) introduced in Section 2.3:

1Note that λ1 and λ2 can be determined from θ using equations (1), (2) and (14). The remaining variables in equation (12),τ1, τ2 and εf1, are exogenously determined.

17

Definition 5

I(θt−1, ηt−1, θt+1

)=

θ if U1 (θ, θt+1, η (θ)) = U2(θ, θt−1, ηt−1

)for some θ ∈ [0, 1]

0 if U1 (θ, θt+1, η (θ)) < U2(θ, θt−1, ηt−1

)for each θ ∈ [0, 1]

1 if U1 (θ, θt+1, η (θ)) > U2(θ, θt−1, ηt−1

)for each θ ∈ [0, 1]

Using the definition of I(θt−1, ηt−1, θt+1

), we can provide a characterisation of equilibria as follows.

Proposition 4 Given initial values θ0 and η0, a sequence {θτ}∞τ=1 constitutes a Perfect Bayesian Equilib-

rium if and only if, we have θt = I(θt−1, ηt−1, θt+1

), where ηt = η (θt−1), for t = 1, 2, ....

There exists a steady-state equilibrium at θ ∈ [0, 1] if and only if θ = I (θ, η (θ) , θ). In other words, if the

proportion of men seeking young brides, and the proportion of young women who decline offers of marriage,

are equal to θ and η (θ) respectively in some period, and θ = I (θ, η (θ) , θ), then there exists an equilibrium

where these proportions remain the same in all future periods.

5.2 Formal Analysis

We are now in a position to investigate, formally, how improving access to schooling impacts upon the

marriage market equilibrium. It is reasonable to represent improved access to schooling as an increase in

the common component in the utility of singlehood from y to, say, y′. For ease of comparison, suppose that

the marriage market is initially in a steady-state equilibrium; i.e. a fixed proportion of men, θs seek young

brides in each period, where θs = I (θs, η (y, θs) , θs). Let us denote by v the first period of the intervention.

Note that, in the first period following the intervention, the reputation of older women on the marriage

market is unaffected by the intervention itself (assuming that the intervention was unanticipated); as their

reputation is determined by the proportion of men seeking young brides, and the proportion of young women

who turned down offers of marriage, in the preceding period. Therefore, for any value of θv, the expected

utility to men from seeking older brides is the same as it would have been in the absence of the intervention:

U2 (θv, θs, η (y, θs)).

On the other hand, the expected utility from seeking a young bride depends on the proportion of young

women who turn down offers of marriage in the current period. As a result of improved access to secondary

schooling, a higher proportion of women will turn down offers of marriage in period v for any expectation

of θv+1: η (y′, θv+1) > η (y, θv+1). This would make it less attractive, given any pair θv, θv+1, for men

to seek young brides (as doing so will be less likely to lead to a marriage): U1 (θv, θv+1, η (y′, θv+1)) <

U1 (θv, θv+1, η (y, θv+1)).

There are, potentially, multiple Perfect Bayesian equilibria following the intervention, each corresponding

to a different value for θv+1. The intervention itself provides no guidance about which of these Perfect

Bayesian equilibria may be realised. Therefore, we focus instead on the ‘Naive Expectations’equilibrium

which, as per Propositon 3, is uniquely determined by θs. In this case, θv solves the equation

U1 (θv, θs, η (y′, θs)) = U2 (θv, θs, η (y, θs)) (15)

It follows that θv < θs.2

2To see this, note thatU1(θs, θs, η

(y′, θs

))< U1 (θs, θs, η (y, θs))

18

Then, in period v + 1, the reputation of older women improve for two reasons: (i) a lower proportion

of them received offers of marriage when young than the preceding cohort (θv < θs) and, (ii) compared to

the preceding cohort, a larger proportion of those who received offers of marriage when young refused them

(η (y′, θs) > η (y, θs)). Therefore, in the second period of the intervention, the expected utility to men from

seeking older brides will be higher, for any value of θv+1, than in period v:

U2 (θv+1, θv, η (y′, θv)) > U2 (θv+1, θs, η (y, θs))

=⇒ U2 (θv, θv, η (y′, θv)) > U1 (θv, θs, η (y

′, θs)) using (15)

=⇒ U2 (θv, θv, η (y′, θv)) > U1 (θv, θv, η (y

′, θv)) using Lemma 1(ii)

Therefore, θv+1 < θv.

Then, in period v + 2, the reputation of older women improves once again as (i) a lower proportion of

them received offers of marriage when young than the preceding cohort (θv+1 < θv) and, (ii) compared to

the preceding cohort, a larger proportion of those who received offers of marriage when young refused them

(η (y′, θv) > η (y′, θs)). Thus, the cycle continues.

Therefore, even if an intervention initially affects the outside options of only a subset of young women,

it can initiate a virtuous cycle in the marriage market such that more and more women opt not to marry

when they are young. The marriage market will reach a new steady state at θ′s, which satisfies the equation

θ′s = I(θ′s, η

(y′, θ′s

), θ′s)

It is straightforward to show that θ′s < θs.

6 Discussion

In recent years, international donors and development agencies have designed and implemented a range of

interventions aimed at expanding the economic and educational opportunities of adolescents, particularly

girls, in developing countries; with the reduction of the incidence of child marriage and postponement of

childbirth among their intended goals.

In this context, it is important to realise that decisions regarding the timing of marriage of a son or

daughter, although they are made within a single household, may be influenced by the choices made by

other households in the community or region. In this paper, we investigated a particular mechanism through

which these individual choices can influence the costs and benefits of early marriage versus late marriage for

everyone in a marriage market: if some qualities of prospective brides are not fully observable at the time

of marriage, then the incidence of child marriage will influence the perceived quality of older women on the

marriage market.

In this situation, expanding opportunities for some adolescent girls — to the extent that they or their

parents choose to postpone their marriage —can trigger a virtuous cycle whereby the perceived quality of

Since U1 (.) is decreasing in its first argument and U2 (.) is increasing in its first argument, it follows that θv < θs.

19

older brides improve, which in turn makes it more attractive for other girls to postpone marriage, and so on.

In a study of the timing of marriage and childbearing in rural Bangladesh (Schuler et al., 2006), this idea is

succinctly captured in a reported interview with an 18-year-old girl:

‘... when asked why her parents had delayed her marriage while her younger sisters had

married at ages 12-15 ... [she replied] "My father thought it was unnecessary for girls to read and

write but in my case he did not object ... None of my peers were sitting idle at home so I also

went to school. Now it is better for girls. They don’t have to pay school fees —the government

finances it ... Everyone has had some schooling, at least up to the eighth or ninth grade. No one

would want to marry an illiterate girl so they are sent to school." p. 2831

The fact that an intervention targeted at adolescent girls can make it more and more attractive for

future cohorts to postpone marriage means that the long-term impact of such interventions on marriage and

subsequent life choices may well exceed the immediate impact.

7 Appendix

Proof. of Lemma 1: (i) By Assumption 3, µ(nf1θtnm

)is strictly decreasing in θt. Therefore, using the

definitions of U1 (θt, θt+1) and U2 (θt, θt−1) in (7) and (8), we have U1 (θt, θt+1) strictly decreasing in θt and

U2 (θt, θt−1) is strictly increasing in θt.

(ii) Using (6), we see that the outside option of young women in period t, vf1, is increasing in the value

of λ2 in period t + 1. Using (2), λ2 is decreasing in θt+1. Therefore, vf1 is decreasing in θt+1.Then, using

(9) and Assumption 5, τ1 is increasing in vf1. It follows from (7) and Assumption 1 that U1 is increasing in

θt+1.

(iii) Using (3), εf2 in period t is increasing in λ1 in period t − 1. Using (1), λ1 is increasing in θt−1.Therefore, εf2 is increasing in θt−1. Then, using (8) and (10) and Assumption 8, U2 is decreasing in θt−1.

Proof. of Lemma 2: Define J (θ) = I (θ, θ)− θ. Given that um (.), µ (.) and ξ (.) are continuous functions,so is J (.). By construction, J (θ) ≥ 0 at θ = 0 and J (θ) ≤ 0 at θ = 1. If J (0) = 0 or J (1) = 0, then thecorresponding θ value would constitute a steady-state. If J (0) > 0 and J (1) < 0, then it follows from the

Intermediate Value Theorem that there exists some θ ∈ [0, 1] such that J (θ) = 0. Therefore, I (θ, θ) = θ.

Proof. of Proposition 2: (i) Consider a monotonic increasing sequence {θt}∞t=0 where θt ∈ [0, 1] for all t.First, we show that this sequence has a point of convergence in the interval (θ0, 1]. Let x = sup {θt}. Bydefinition, for each ε > 0, ∃T ∈ N such that |θT − x| < ε (if not, x would not be the supremum of the

sequence). Given that {θt}∞t=0 is a monotonic sequence, it follows that |x− θt| < ε for t ≥ T .Therefore, for r, s ≥ T , we obtain

|θr − θs| ≤ |x− θr|+ |x− θs|

≤ 2ε

Therefore, {θt}∞t=0 constitutes a Cauchy sequence. Every Cauchy sequence of real numbers has a limit

20

(see, for example, Lang, p.143). Therefore, {θt}∞t=0 has a limit. Let y = limt→∞ {θt}. It follows thaty = sup (θt) = x.

Next, we show that y corresponds to a steady-state equilibrium, i.e. I (y, y) = y. We prove this by

contradiction. Suppose I (y, y) > y. Then, as I (.) is continuous in both arguments, we can find ε suffi ciently

small such that

I (y − ε, y − ε) > y

Given the convergence property of {θt}∞t=0, there exists T suffi ciently large such that for t ≥ T , we have

θt > y − ε. Therefore,I (θT , θT+2) > y

=⇒ θT+1 > y

This contradicts the earlier statement that y is the supremum of the sequence.

Now suppose that I (y, y) < y and let z = y − I (y, y) > 0. Consider some δ > 0. By Lemma 1 and

continuity of I (.), we have, for each θ1, θ2 ∈ (y − δ, y)

I (θ1, θ2) ≤ y − z

Given the convergence property of {θt}∞t=0, there exists T suffi ciently large such that for t ≥ T , we have

θt ∈ (y − δ, y). Therefore for each r ≥ T + 1, we have

θr = I (θr−1, θr+1) ≤ y − z

Therefore,

sup {θt}∞t=T+1 ≤ y − z

Given that the sequence {θt}∞t=0 is monotonic, we have

sup {θt}∞t=0 ≤ y − z < y

which contradicts the earlier statement that y is the supremum of the sequence. Therefore, it must be that

y corresponds to a steady-state equilibrium.

(ii) The proof for a monotonic decreasing sequence proceeds in the same manner as above, such that

we let x = inf {θt} and show that the sequence converges to x. Using the reasoning by contradiction usedabove, we then show that x is a steady-state value, i.e. I (x, x) = x.

Proof. of Proposition 3: Given θ0, we can construct a sequence {θτ}∞τ=1 as follows. Let θt+1 = I (θt, θt) for

t = 1, 2, ... By construction, the sequence {θτ}∞τ=1, together with the initial value θ0, satisfies the definitionof a ‘Naive Expectations’equilibrium in 4. Furthermore, as I (., .) is increasing in both arguments by Lemma

1,given θ0, no other sequence{θ′τ}∞τ=1

can satisfy the definition in 4. Therefore, given the initial value θ0,

there is a unique sequence that satisfies the criteria of the ‘Naive Expectations’Equilibrium.

(i) Suppose I (θ0, θ0) > θ0. As I (1, 1) ≤ 1, and I (., .) is a continuous function, it follows that there

exists at least one steady-state value in the interval (θ0, 1]. Let us denote the first (i.e. smallest) of these

steady-state values as θs; i.e. I (θs, θs) = θs. By continuity of I (., .), we have θ < I (θ, θ) for θ ∈ [θ0, θs).

21

By construction, θ1 = I (θ0, θ0). Since θ0 < I (θ0, θ0), we have θ1 > θ0. We can show by contradiction

that θ1 < θs. Define K (θ) = I (θ, θ). By Lemma 1, K (θ) is increasing in θ. Therefore, K−1 (.) is an

increasing function. Therefore, if θ1 > θs, then K−1 (θ1) > K−1 (θs). By construction, K (θ0) = I (θ0, θ0) =

θ1. Therefore K−1 (θ0) = θ1. Therefore θ0 > θs, which is a contradiction of our initial premise.

By the same reasoning, θ2, θ3... < θs. Since I (θ, θ) > θ for θ < θs, it follows that {θτ}∞τ=1 is an increasingsequence. Following the proof of Proposition 2, we can show that the sequence converges to y = sup {θτ}and that y corresponds to a steady-state. As all elements of the sequence are smaller than θs, it follows that

the steady-state must be θs.

(ii) Suppose I (θ0, θ0) < θ0, As I (0, 0) ≥ 1 and I (., .) is a continuous function, it follows that there is atleast one steady-state value in the interval [0, θ0). Let us denote the largest of these steady-state values as

θ′s; i.e. I(θ′s, θ

′s

)= θ′s. Following the reasoning in part (i) above, we can show that sequence{θτ}

∞τ=1 is a

decreasing sequence that converges to θ′s.

(iii) Suppose I (θ0, θ0) = θ0. Since θt+1 = I (θt, θt) for t = 1, 2, .., it follows that θ1 = I (θ0, θ0) = θ0, and

θ2 = I (θ1, θ1) = I (θ0, θ0) = θ0, etc. Therefore, each element of the sequence {θτ}∞τ=1 is equal to θ0.

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